Difference between Prim and Dijkstra graph algorithm

In both algorithms queue Q contains all vertices that are not 'done' yet, i.e. white and gray according to common terminology (see http://www.cs.cornell.edu/courses/cs2112/2014fa/lectures/lecture.html?id=ssp). In Dijkstra's algorithm, the black vertex cannot be relaxed, because if it could, that would mean that its distance was not correct beforehand (contradicts with property of black nodes). So there is no difference whether you check v in Q or not. In Prim's algorithm, it is possible to find an edge of small weight, that leads to already black vertex. That's why if we do not check v in Q, then the value in vertex v can change indeed. Normally, it does not matter, because we never read min-weight value for black vertices. However, your pseudocode is using https://en.wikipedia.org/wiki/Binary_heap data structure. In this case each modification of vertex values must be accompanied with DecreaseKey. Clearly, it is not valid to call DecreaseKey for black vertices, because they are not in heap. That's why I suppose author decided to check for v in Q explicitly. Speaking generally, the codes for Dijkstra's and Prim's algorithms are usually absolutely same, except for a minor difference: Prim's algorithm checks w(u, v) for being less than D(v) in RELAX. Dijkstra's algorithm checks D(u) + w(u, v) for being less D(v) in RELAX.